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Unformatted text preview: MA3252 LINEAR AND NETWORK OPTIMIZATION Topic 5: Sensitivity Analysis 1 / 27 Sensitivity (or post optimality) analysis deals with the study of possible changes in the optimal solution as a result of making changes in the original problem. Why study sensitivity analysis? 1 In practice, there is often incomplete knowledge of the problem data. We cannot predict changes of data, but we may wish to predict the effects of certain data changes, e.g. to which parameters the profit (or cost) is more (or less) sensitive. 2 We may be able to change some input parameters. Which parameters are worth change and how much should they be changed, allowing for offset of costs? 2 / 27 How to analyze sensitivity? Consider the standard form primal problem and its dual problem: ( P ) min c x s.t. Ax = b x ≥ ( D ) max p b s.t. p A ≤ c . We shall study the dependence of the optimal objective value and the optimal solution on the coefficient matrix A , the right-hand side vector b , and the cost vector c . 3 / 27 Available information: Suppose x * is a optimal primal basic feasible solution, with associated optimal basis B . Then x * B = B- 1 b ≥ and the optimal cost is c x * = c B x * B = c B B- 1 b . Given changes of A , b , c , we look for a new optimal solution. 1 First, we check if the current optimal basis B and solution x * is still optimal. 2 If not, we compute a new optimal solution, starting from x * and B . Key conditions that we need to check are: B- 1 b ≥ (Feasibility) c- c B B- 1 A ≥ (Optimality) 4 / 27 Sensitivity Analysis We shall look for ranges of parameter changes under which current basis is still optimal. If the change is beyond this range, we look for algorithm that finds a new optimal solution without having to solve the new problem from scratch. 5 / 27 Changes in right-hand side vector b Suppose that some component b i of the requirement vector b is changed to b i + δ , i.e. b is changed to b + δ e i where e i is the i th unit vector. Optimality conditions are unaffected by the change in b . It remains to examine the feasibility condition B- 1 ( b + δ e i ) ≥ , i.e. x * B + δ ( B- 1 e i ) ≥ . This provides a range for δ to maintain feasibility. However, if δ is not in the range, then the feasibility condition is violated, and we apply dual simplex method starting from the basis B . 6 / 27 Changes in cost vector c Suppose that some component c j of the cost vector c is changed to c j + δ . The primal feasibility condition is not affected by the change of c . It thus remains to examine the optimality condition c- c B B- 1 A ≥ ....
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